Organizers: the course is organized in partnership with three PhD Schools: 

• PhD  on Mathematics, Dept. Mathematics & Physics, University Roma Tre;

• PhD on Theoretical & Applied Mechanics, Dept. Aeronautical & Mechanical Engineering, Sapienza University, Roma;

• PhD on Structural & Geotechnical Engineering, Dept. of Structural and Geotechnical Engineering, Sapienza University, Roma.

Schedule: June 2023;

Four lectures: Mon 19, Wed 21, Fri 23, Mon 26, from 9:00 to 13:00 (break included).

Venue: Lecture Room 36, first floor of building D (aka, DICEA building), at Engineering School of Sapienza University, via Eudossiana 18, Rome.

Teaching Modalities: all lectures will be in hybrid mode (in person, streaming, playback); streaming and playback with M-Teams. Due to the "hands-on" approach, participants are recommended to attend in person.

Goal: understand the fundamentals of continuum mechanics through worked examples, with a focus on the relations between mechanics and differential geometry. Participants will tackle some typical problems of continuum mechanics, and will learn to implement a given problem using the weak formulation into the COMSOL software and to discuss the solution.

Software: a trial license for the latest COMSOL version can be dowloaded at the following link: Get your trial license for COMSOL 

Synopsis of lectures (tentative program)

1) Browse a model of nonlinear solid mechanics, from the implementation to the solution.

A first glance at the fundamentals of continuum mechanics: Kinematics, Constitutive, and Balance laws.

Differential form (strong) versus Integral form (weak).

What is a test function (aka, virtual displacement or virtual velocity)

Worked example: large deformations of a hyperelastic solid under loadings.

2) Material Versus Spatial description.

A continuum body as a differentiable manifold.

Tell the difference between tensors: strain tensor versus stress tensor.

Geometric elements; change of densities.

Pull back & push forward of scalar, vector and tensor fields.

3) Solid mechanics versus Fluid mechanics

Kinematical constraints; isochoric motion.

Reference stress (Piola) & Actual stress (Cauchy).

Polar decomposition theorem; eigen-space of the stress tensor and of the strain tensor

4) Non linear solid mechanics and the Elastic Metric

Worked example: large deformations of a hyperelastic solid under distortions. The notion of Elastic metric and Target metric.

5) Material response

Worked example: from elastic energy  to the constitutive law for the stress.

Transversely isotropic materials. Fiber reinforced materials.

Worked example: fiber reinforced hyperelastic solid under traction.

6) Fluid dynamics

Tackling Navier-Stokes equations.

Worked examples: fluid in a channel; fluid around an obstacle.

7) Fluid-Structure interactions - theory

Worked examples: understand the moving mesh technique; how to write the problem of a beam immersed in a fluid.

8) Fluid-Structure interactions - practice

Worked example: implement and solve the problem of an oscillating beam immersed in a fluid.